3.3 Theoretical Aspects

# 3.3 Theoretical Aspects - 3.3 3.3 Systems of Linear...

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3.3. 3.3. Systems of Linear equations - Theoretical aspects 1. (a) F p.170, Example 1 (c) (b) F (c) T Any homogeneous system has at least one solution, namely, the zero vector (d) F p.174, Theorem 3.10 (e) F (f) F p.172, Theorem 3.9 (g) T If A is invertible, then AX = 0 has no nonzero solutions (h) T 2. Let K be the solution set of the given system and A is the coeﬃcient ma- trix of the system (a) A = ± 1 3 2 6 ˆ ± 1 3 0 0 rank ( A ) = 1 , dim ( K ) = 2 - 1 = 1 ‰± - 3 1 ¶² : a basis for K 172 PNU-MATH

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3.3. (b) A = ± 1 1 - 1 4 1 - 2 ˆ ± 1 0 0 0 1 0 rank ( A ) = 2 , dim ( K ) = 3 - 2 = 1 Since (1 , 2 , 3) t is a solution to AX = 0, 1 2 3 : a basis for K (c) A = ± 1 2 - 1 2 1 1 ˆ ± 1 0 0 0 1 0 rank ( A ) = 2 , dim ( K ) = 3 - 2 = 1 Since ( - 1 , 1 , 1) t is a solution to AX = 0, - 1 1 1 : a basis for K (d) A = 2 1 - 1 1 - 1 1 1 2 - 2 ˆ 1 0 0 0 1 0 0 0 0 rank ( A ) = 2 , dim ( K
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## This note was uploaded on 02/14/2012 for the course MAS 4105 taught by Professor Rudyak during the Spring '09 term at University of Florida.

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3.3 Theoretical Aspects - 3.3 3.3 Systems of Linear...

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